Vibrational effects in charge transport through a molecular double quantum dot

Recent progress in the field of molecular electronics has revealed the fundamental importance of the coupling between the electronic degrees of freedom and specific vibrational modes. Considering the examples of a molecular dimer and a carbon nanotube double quantum dot, we here theoretically investigate transport through a two-site system that is strongly coupled to a single vibrational mode. Using a quantum master equation approach, we demonstrate that, depending on the relative positions of the two dots, electron-phonon interactions can lead to negative differential conductance and suppression of the current through the system. We also discuss the experimental relevance of the presented results and possible implementations of the studied system.

Figure 1

Schematic illustrations of molecular double quantum dot systems discussed. Both sites are coupled to the same vibrational mode as well as to respective electrodes. Exemplary phonon modes are schematically depicted above. (a) A carbon nanotube with a pair of quantum dots (in purple) formed by applying a gate potential (as labeled). (b) A single-molecule junction based on a zinc-porphyrin dimer bridging a pair of graphene electrodes. Each of the porphyrin sites acts as a quantum dot.

Figure 2

Schematic illustration of a double quantum dot studied in this work. The two sites are coupled to each other with the strength $J$. The rates of electron hopping between the electrodes and respective sites are given by ${\gamma }_L,{\gamma }_R$—see text. Note that our model may apply to either a transverse or a longitudinal vibrational mode (as sketched in Fig. 1), provided the coupling strength of charge state to a single mode dominates.

Figure 3

Current-voltage characteristics and voltage-dependent populations of the sites in a strongly coupled two-site molecular system. The calculation was performed for a symmetric system: ${\overline{\varepsilon }}_L={\overline{\varepsilon }}_R=0.05$ eV. The energy of the vibrational mode was taken to be $\omega =0.02$ eV, the electron-phonon coupling constants $|g_L|=|g_R|=0.018$ eV, the coupling between the sites $J=0.3$ eV [90], and between the sites and the leads as ${\gamma }_L={\gamma }_R=0.001$ eV. The temperature was assumed to be 10 K leading to appreciable thermal broadening.

Figure 4

Current at bias voltage $V_b=2{\overline{\varepsilon }}_L+\omega$ (first step in the current-voltage characteristics). The calculation was performed for a symmetric system: ${\overline{\varepsilon }}_L={\overline{\varepsilon }}_R=0.05$ eV. The energy of the vibrational mode was taken to be $\omega =0.02$ eV, the electron-phonon coupling constants $|g_L|=|g_R|=0.018$ eV, the coupling between the sites and the leads is ${\gamma }_L={\gamma }_R=0.001$ eV, and the temperature was assumed to be 10 K.

Figure 5

Current-voltage characteristics and voltage-dependent populations of the sites in a strongly coupled two-site molecular system. The model is identical to the one used in Fig. 3 except for the coupling between the sites $J=0.01$ eV, and between the sites and the leads ${\gamma }_L={\gamma }_R=0.05$ eV.

Figure 6

Current-voltage characteristics (a) and current at bias voltage $V_b=2{\overline{\varepsilon }}_L+\omega$ (b) for a symmetric DQD: ${\overline{\varepsilon }}_L={\overline{\varepsilon }}_R=0.05$. The energy of the vibrational mode was taken to be $\omega =0.02$ eV, the electron-phonon coupling constants $|g_L|=|g_R|=\lambda \omega$, the intersite coupling $J=0.01$ eV, and between the sites and the leads ${\gamma }_L={\gamma }_R=0.05$ eV, $T=10$ K.

Figure 7

Current-voltage characteristics for a detuned DQD: ${\overline{\varepsilon }}_L=0.05$ eV, ${\overline{\varepsilon }}_R=0.15$ eV. The energy of the vibrational mode was taken to be $\omega =0.02$ eV, the electron-phonon coupling constants $|g_L|=|g_R|=0.018$ eV, the intersite coupling $J=0.01$ eV, and between the sites and the leads ${\gamma }_L={\gamma }_R=0.05$ eV, $T=10$ K.

Figure 8

Square moduli of diagonal matrix elements of $X_L^{†}{X}_R$ operators for different values of $\lambda$ as used in Fig. 6.